Algebra 2
The quadratic formula
For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) / (2a). The discriminant b² − 4ac tells you how many real solutions exist before you compute them.
Why it matters
The formula always works, so it is the safety net when factoring fails. Understanding the discriminant also explains why some models have no real answer — a common source of confusion in physics problems.
Worked example
Solve 2x² − 4x − 3 = 0
- 1Identify coefficients. a = 2, b = −4, c = −3
- 2Compute the discriminant. (−4)² − 4(2)(−3) = 16 + 24 = 40
- 3Substitute. x = (4 ± √40) / 4
- 4Simplify the surd. √40 = 2√10, so x = (4 ± 2√10)/4 = (2 ± √10)/2
Answer: x = (2 + √10)/2 or x = (2 − √10)/2
Common mistakes
- Losing the sign of b when it is negative — remember −b means −(−4) = 4.
- Dividing only one term by 2a.
- Leaving the surd unsimplified when the question asks for exact form.
How to check your answer
- A positive discriminant means two real solutions, zero means one, negative means none.
- Add your two roots: they should equal −b/a.
Related topics
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